Let be the space of base-point-preserving maps from a connected finite CW complex to a connected space . Consider a CW complex of the form and a space whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map is greater than the Whitehead length of , then has the rational homotopy type of the product space . This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex are greater than and the connectivity of is greater than or equal to , then the mapping space can be decomposed rationally as the product of iterated loop spaces.
"A rational splitting of a based mapping space." Algebr. Geom. Topol. 6 (1) 309 - 327, 2006. https://doi.org/10.2140/agt.2006.6.309